Let p be a random probability measure chosen by a Dirichlet process whose parameter a is a finite measure with support contained in $[0, +\infty)$ and suppose that $V = \int x^2p(dx)-[\int xp(dx)]^2$ is a (finite)random variable. This paper deals with the distribution of $V$, which is given in a rather general case. A simple application to Bayesian bootstrap is also illustrated.
"Some new results for Dirichlet priors." Ann. Statist. 28 (5) 1390 - 1413, October2000. https://doi.org/10.1214/aos/1015957399